You are given that $3^{400}\equiv 1\pmod{1000}$. What are the last three digits of $3^{12000}$?
Answer: The last three digits are the same as the remainder when divided by $1000$.

$3^{400}\equiv 1\pmod{1000}\implies 3^{12000}=(3^{400})^{30}\equiv 1^{30}=1\pmod{1000}$.

Thus, the last three digits are $\boxed{001}$.